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Signal Processing Week 2 Lecture A:

Continuous Time Fourier Series (Trig Form)

Transforms

  • Signals are functions that convey information
  • Straightfoward ways (like amplitude vs time) of representing a signal may not be the best for showing its properties
  • Example: look at frequency over time instead of pressure over time
  • Transform: alternate view of a signal
  • 6.003: family of transforms known as fourier transforms, where signals are represented as sums of harmonically-related sinusoids

f(t)=c0+k=1(ckcos(kω0t)+dksin(kω0t))

  • We already dealt with this representation in PSET #1

Key Terms:

  • A periodic signal repats after some amount of time T where

x(t)=x(t+T)

  • A function periodic in T is periodic in nT for all integer n
  • The smallest value T for which this holds is called the fundamental period
  • The frequency ω0=2πT is the signal's fundamental frequency
  • Frequencies are harmonically related if they are integer multiples of some T

Sinusoids Note:

Acos(ωt)+Bsin(ωt)=(A2+B2)cos(ωt+tan1(ba))

  • Can be proven through converting to complex exponential form and taking the real part
  • Demonstrates that the sum of sine and cosine terms yields another sinusoid

Integrating over a period

t0t0+Tsin(2πkTt)dt=0
t0t0+Tcos(2πkTt)dt=0

  • Product Identities

t0t0+Tsin(2πkTt)cos(2πmTt)dt=0

  • True for all integers k, m

t0t0+Tcos(2πkTt)cos(2πmTt)dt

  • This expression is equal to T/2 if k=m, otherwise 0
  • Can be proven using trig identities

Fourier Series

  • Fourier Series are sums of harmonically related sinusoids

f(t)=c0+k=1(ckcos(kω0t)+dksin(kω0t))

  • ω0 is the fundamental frequency

  • When can we represent a signal as sums of harmonic components:

  • Only periodic signals, all harmonics of ω0 are periodic in T=2π/ω0

The c0 term

  • The constant c0 is known as the "DC" term
  • Constant offset of a signal
  • Average vvalue of the signal over one period
  • Can be computed by computing the definite integral over a T f is periodic over and dividing by T

Computing the other ck terms

f(t)=c1cos(ω0t)+d1sin(ω0t)+c3cos(3ω0t)+d5sin(5ω0t)

  • Stratgy for finding c3: multiply by c3 and integrate the result over one period
  • Works because all terms in the integral will integrate to 0 by the trig rules from earlier, with the exception of the c3cos(3ω0t) term

  • Rules:

ck=2Tt0t0+Tf(t)cos(2πktT)dt
dk=2Tt0t0+Tf(t)sin(2πktT)dt

  • Why do we need both sin and cos components?
  • Allows for representation of both even and odd components of the signal

Practice

f(t)=0.8sin(6πt)0.3cos(6πt)+0.75cos(12πt)

  • ω0=6π

  • T=2πω0

  • T=1/3

What are the Fourier series coefficients for f(t)?

  • c0=0
  • c1=0.3
  • c2=0.75
  • d1=0.8

Analyzing a Square Wave

Which are true:

  • ck=0 for all k
  • True! The function is odd, so it can be analyzed with only sin terms
  • dk=0 if k is even
  • True (can be proven with integration rules)
  • |dk| decreases with k
  • True
  • There are an infinite number of non-zero dk -True

1 term

2 terms

10 terms

100 terms

Why does the corner distort?

  • With finitely many terms, there are 'overshoots' or 'rings' at discontinuities
  • Known as Gibb's Phenomenon
  • You can decrease the ringing by descreasing the magnitudes of Fourier coefficients at higher frequencies

Triangular Wave

  • 6 terms
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4.61633746e-02+2.08671610e-02j, 4.64222025e-02+2.02848387e-02j,\n 4.66736851e-02+1.96993067e-02j, 4.69177826e-02+1.91106578e-02j,\n 4.71544564e-02+1.85189850e-02j, 4.73836691e-02+1.79243820e-02j,\n 4.76053843e-02+1.73269429e-02j, 4.78195670e-02+1.67267621e-02j,\n 4.80261834e-02+1.61239348e-02j, 4.82252007e-02+1.55185561e-02j,\n 4.84165874e-02+1.49107220e-02j, 4.86003132e-02+1.43005286e-02j,\n 4.87763491e-02+1.36880725e-02j, 4.89446673e-02+1.30734505e-02j,\n 4.91052410e-02+1.24567600e-02j, 4.92580449e-02+1.18380984e-02j,\n 4.94030548e-02+1.12175637e-02j, 4.95402478e-02+1.05952541e-02j,\n 4.96696022e-02+9.97126802e-03j, 4.97910974e-02+9.34570421e-03j,\n 4.99047142e-02+8.71866165e-03j, 5.00104348e-02+8.09023955e-03j,\n 5.01082423e-02+7.46053735e-03j, 5.01981213e-02+6.82965469e-03j,\n 5.02800575e-02+6.19769139e-03j, 5.03540381e-02+5.56474743e-03j,\n 5.04200512e-02+4.93092298e-03j, 5.04780865e-02+4.29631832e-03j,\n 5.05281347e-02+3.66103387e-03j, 5.05701880e-02+3.02517013e-03j,\n 5.06042396e-02+2.38882773e-03j, 5.06302843e-02+1.75210735e-03j,\n 5.06483178e-02+1.11510974e-03j, 5.06583373e-02+4.77935682e-04j,\n 5.06603413e-02-1.59313995e-04j, 5.06543294e-02-7.96538464e-04j,\n 5.06403026e-02-1.43363690e-03j, 5.06182631e-02-2.07050849e-03j,\n 5.05882143e-02-2.70705247e-03j, 5.05501611e-02-3.34316812e-03j,\n 5.05041095e-02-3.97875479e-03j, 5.04500667e-02-4.61371191e-03j,\n 5.03880412e-02-5.24793901e-03j, 5.03180430e-02-5.88133574e-03j,\n 5.02400831e-02-6.51380187e-03j, 5.01541738e-02-7.14523735e-03j,\n 5.00603287e-02-7.77554224e-03j, 4.99585626e-02-8.40461683e-03j,\n 4.98488917e-02-9.03236158e-03j, 4.97313334e-02-9.65867715e-03j,\n 4.96059061e-02-1.02834645e-02j, 4.94726298e-02-1.09066246e-02j,\n 4.93315256e-02-1.15280591e-02j, 4.91826157e-02-1.21476694e-02j,\n 4.90259238e-02-1.27653577e-02j, 4.88614746e-02-1.33810262e-02j,\n 4.86892942e-02-1.39945773e-02j, 4.85094097e-02-1.46059142e-02j,\n 4.83218498e-02-1.52149400e-02j, 4.81266439e-02-1.58215584e-02j,\n 4.79238231e-02-1.64256733e-02j, 4.77134194e-02-1.70271893e-02j,\n 4.74954661e-02-1.76260111e-02j, 4.72699977e-02-1.82220439e-02j,\n 4.70370498e-02-1.88151935e-02j, 4.67966594e-02-1.94053661e-02j,\n 4.65488645e-02-1.99924681e-02j, 4.62937042e-02-2.05764068e-02j,\n 4.60312189e-02-2.11570897e-02j, 4.57614503e-02-2.17344250e-02j,\n 4.54844409e-02-2.23083213e-02j, 4.52002346e-02-2.28786878e-02j,\n 4.49088763e-02-2.34454343e-02j, 4.46104123e-02-2.40084710e-02j,\n 4.43048896e-02-2.45677089e-02j, 4.39923566e-02-2.51230596e-02j,\n 4.36728628e-02-2.56744350e-02j, 4.33464588e-02-2.62217481e-02j,\n 4.30131962e-02-2.67649121e-02j, 4.26731276e-02-2.73038412e-02j,\n 4.23263070e-02-2.78384500e-02j, 4.19727892e-02-2.83686541e-02j,\n 4.16126301e-02-2.88943694e-02j, 4.12458867e-02-2.94155128e-02j,\n 4.08726171e-02-2.99320019e-02j, 4.04928803e-02-3.04437549e-02j,\n 4.01067364e-02-3.09506908e-02j, 3.97142465e-02-3.14527295e-02j,\n 3.93154726e-02-3.19497915e-02j, 3.89104780e-02-3.24417981e-02j,\n 3.84993267e-02-3.29286715e-02j, 3.80820837e-02-3.34103347e-02j,\n 3.76588150e-02-3.38867115e-02j, 3.72295877e-02-3.43577264e-02j,\n 3.67944696e-02-3.48233050e-02j, 3.63535296e-02-3.52833736e-02j,\n 3.59068374e-02-3.57378593e-02j, 3.54544638e-02-3.61866903e-02j,\n 3.49964803e-02-3.66297956e-02j, 3.45329594e-02-3.70671050e-02j,\n 3.40639745e-02-3.74985494e-02j, 3.35895996e-02-3.79240605e-02j,\n 3.31099099e-02-3.83435709e-02j, 3.26249813e-02-3.87570143e-02j,\n 3.21348906e-02-3.91643252e-02j, 3.16397152e-02-3.95654393e-02j,\n 3.11395335e-02-3.99602930e-02j, 3.06344247e-02-4.03488239e-02j,\n 3.01244686e-02-4.07309704e-02j, 2.96097461e-02-4.11066722e-02j,\n 2.90903384e-02-4.14758698e-02j, 2.85663278e-02-4.18385047e-02j,\n 2.80377973e-02-4.21945196e-02j, 2.75048303e-02-4.25438582e-02j,\n 2.69675114e-02-4.28864651e-02j, 2.64259254e-02-4.32222862e-02j,\n 2.58801582e-02-4.35512684e-02j, 2.53302959e-02-4.38733595e-02j,\n 2.47764257e-02-4.41885086e-02j, 2.42186352e-02-4.44966659e-02j,\n 2.36570126e-02-4.47977825e-02j, 2.30916468e-02-4.50918109e-02j,\n 2.25226273e-02-4.53787045e-02j, 2.19500440e-02-4.56584180e-02j,\n 2.13739877e-02-4.59309070e-02j, 2.07945494e-02-4.61961284e-02j,\n 2.02118208e-02-4.64540403e-02j, 1.96258942e-02-4.67046019e-02j,\n 1.90368621e-02-4.69477736e-02j, 1.84448180e-02-4.71835168e-02j,\n 1.78498553e-02-4.74117942e-02j, 1.72520683e-02-4.76325698e-02j,\n 1.66515515e-02-4.78458086e-02j, 1.60484000e-02-4.80514768e-02j,\n 1.54427092e-02-4.82495419e-02j, 1.48345749e-02-4.84399727e-02j,\n 1.42240934e-02-4.86227388e-02j, 1.36113612e-02-4.87978115e-02j,\n 1.29964753e-02-4.89651631e-02j, 1.23795330e-02-4.91247669e-02j,\n 1.17606320e-02-4.92765979e-02j, 1.11398701e-02-4.94206319e-02j,\n 1.05173455e-02-4.95568462e-02j, 9.89315680e-03-4.96852193e-02j,\n 9.26740273e-03-4.98057307e-02j, 8.64018229e-03-4.99183615e-02j,\n 8.01159474e-03-5.00230938e-02j, 7.38173953e-03-5.01199111e-02j,\n 6.75071633e-03-5.02087980e-02j, 6.11862497e-03-5.02897404e-02j,\n 5.48556548e-03-5.03627257e-02j, 4.85163801e-03-5.04277421e-02j,\n 4.21694288e-03-5.04847795e-02j, 3.58158052e-03-5.05338287e-02j,\n 2.94565144e-03-5.05748821e-02j, 2.30925629e-03-5.06079332e-02j,\n 1.67249574e-03-5.06329766e-02j, 1.03547056e-03-5.06500085e-02j,\n 3.98281540e-04-5.06590262e-02j, -2.38970500e-04-5.06600282e-02j,\n -8.76184728e-04-5.06530144e-02j, -1.51326032e-03-5.06379858e-02j,\n -2.15009647e-03-5.06149449e-02j, -2.78659242e-03-5.05838953e-02j,\n -3.42264745e-03-5.05448420e-02j, -4.05816092e-03-5.04977910e-02j,\n -4.69303228e-03-5.04427498e-02j, -5.32716107e-03-5.03797273e-02j,\n -5.96044695e-03-5.03087332e-02j, -6.59278972e-03-5.02297789e-02j,\n -7.22408933e-03-5.01428768e-02j, -7.85424589e-03-5.00480408e-02j,\n -8.48315969e-03-4.99452857e-02j, -9.11073122e-03-4.98346279e-02j,\n -9.73686117e-03-4.97160849e-02j, -1.03614505e-02-4.95896754e-02j,\n -1.09844003e-02-4.94554195e-02j, -1.16056121e-02-4.93133383e-02j,\n -1.22249876e-02-4.91634543e-02j, 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1.67249574e-03+5.06329766e-02j,\n 2.30925629e-03+5.06079332e-02j, 2.94565144e-03+5.05748821e-02j,\n 3.58158052e-03+5.05338287e-02j, 4.21694288e-03+5.04847795e-02j,\n 4.85163801e-03+5.04277421e-02j, 5.48556548e-03+5.03627257e-02j,\n 6.11862497e-03+5.02897404e-02j, 6.75071633e-03+5.02087980e-02j,\n 7.38173953e-03+5.01199111e-02j, 8.01159474e-03+5.00230938e-02j,\n 8.64018229e-03+4.99183615e-02j, 9.26740273e-03+4.98057307e-02j,\n 9.89315680e-03+4.96852193e-02j, 1.05173455e-02+4.95568462e-02j,\n 1.11398701e-02+4.94206319e-02j, 1.17606320e-02+4.92765979e-02j,\n 1.23795330e-02+4.91247669e-02j, 1.29964753e-02+4.89651631e-02j,\n 1.36113612e-02+4.87978115e-02j, 1.42240934e-02+4.86227388e-02j,\n 1.48345749e-02+4.84399727e-02j, 1.54427092e-02+4.82495419e-02j,\n 1.60484000e-02+4.80514768e-02j, 1.66515515e-02+4.78458086e-02j,\n 1.72520683e-02+4.76325698e-02j, 1.78498553e-02+4.74117942e-02j,\n 1.84448180e-02+4.71835168e-02j, 1.90368621e-02+4.69477736e-02j,\n 1.96258942e-02+4.67046019e-02j, 2.02118208e-02+4.64540403e-02j,\n 2.07945494e-02+4.61961284e-02j, 2.13739877e-02+4.59309070e-02j,\n 2.19500440e-02+4.56584180e-02j, 2.25226273e-02+4.53787045e-02j,\n 2.30916468e-02+4.50918109e-02j, 2.36570126e-02+4.47977825e-02j,\n 2.42186352e-02+4.44966659e-02j, 2.47764257e-02+4.41885086e-02j,\n 2.53302959e-02+4.38733595e-02j, 2.58801582e-02+4.35512684e-02j,\n 2.64259254e-02+4.32222862e-02j, 2.69675114e-02+4.28864651e-02j,\n 2.75048303e-02+4.25438582e-02j, 2.80377973e-02+4.21945196e-02j,\n 2.85663278e-02+4.18385047e-02j, 2.90903384e-02+4.14758698e-02j,\n 2.96097461e-02+4.11066722e-02j, 3.01244686e-02+4.07309704e-02j,\n 3.06344247e-02+4.03488239e-02j, 3.11395335e-02+3.99602930e-02j,\n 3.16397152e-02+3.95654393e-02j, 3.21348906e-02+3.91643252e-02j,\n 3.26249813e-02+3.87570143e-02j, 3.31099099e-02+3.83435709e-02j,\n 3.35895996e-02+3.79240605e-02j, 3.40639745e-02+3.74985494e-02j,\n 3.45329594e-02+3.70671050e-02j, 3.49964803e-02+3.66297956e-02j,\n 3.54544638e-02+3.61866903e-02j, 3.59068374e-02+3.57378593e-02j,\n 3.63535296e-02+3.52833736e-02j, 3.67944696e-02+3.48233050e-02j,\n 3.72295877e-02+3.43577264e-02j, 3.76588150e-02+3.38867115e-02j,\n 3.80820837e-02+3.34103347e-02j, 3.84993267e-02+3.29286715e-02j,\n 3.89104780e-02+3.24417981e-02j, 3.93154726e-02+3.19497915e-02j,\n 3.97142465e-02+3.14527295e-02j, 4.01067364e-02+3.09506908e-02j,\n 4.04928803e-02+3.04437549e-02j, 4.08726171e-02+2.99320019e-02j,\n 4.12458867e-02+2.94155128e-02j, 4.16126301e-02+2.88943694e-02j,\n 4.19727892e-02+2.83686541e-02j, 4.23263070e-02+2.78384500e-02j,\n 4.26731276e-02+2.73038412e-02j, 4.30131962e-02+2.67649121e-02j,\n 4.33464588e-02+2.62217481e-02j, 4.36728628e-02+2.56744350e-02j,\n 4.39923566e-02+2.51230596e-02j, 4.43048896e-02+2.45677089e-02j,\n 4.46104123e-02+2.40084710e-02j, 4.49088763e-02+2.34454343e-02j,\n 4.52002346e-02+2.28786878e-02j, 4.54844409e-02+2.23083213e-02j,\n 4.57614503e-02+2.17344250e-02j, 4.60312189e-02+2.11570897e-02j,\n 4.62937042e-02+2.05764068e-02j, 4.65488645e-02+1.99924681e-02j,\n 4.67966594e-02+1.94053661e-02j, 4.70370498e-02+1.88151935e-02j,\n 4.72699977e-02+1.82220439e-02j, 4.74954661e-02+1.76260111e-02j,\n 4.77134194e-02+1.70271893e-02j, 4.79238231e-02+1.64256733e-02j,\n 4.81266439e-02+1.58215584e-02j, 4.83218498e-02+1.52149400e-02j,\n 4.85094097e-02+1.46059142e-02j, 4.86892942e-02+1.39945773e-02j,\n 4.88614746e-02+1.33810262e-02j, 4.90259238e-02+1.27653577e-02j,\n 4.91826157e-02+1.21476694e-02j, 4.93315256e-02+1.15280591e-02j,\n 4.94726298e-02+1.09066246e-02j, 4.96059061e-02+1.02834645e-02j,\n 4.97313334e-02+9.65867715e-03j, 4.98488917e-02+9.03236158e-03j,\n 4.99585626e-02+8.40461683e-03j, 5.00603287e-02+7.77554224e-03j,\n 5.01541738e-02+7.14523735e-03j, 5.02400831e-02+6.51380187e-03j,\n 5.03180430e-02+5.88133574e-03j, 5.03880412e-02+5.24793901e-03j,\n 5.04500667e-02+4.61371191e-03j, 5.05041095e-02+3.97875479e-03j,\n 5.05501611e-02+3.34316812e-03j, 5.05882143e-02+2.70705247e-03j,\n 5.06182631e-02+2.07050849e-03j, 5.06403026e-02+1.43363690e-03j,\n 5.06543294e-02+7.96538464e-04j, 5.06603413e-02+1.59313995e-04j,\n 5.06583373e-02-4.77935682e-04j, 5.06483178e-02-1.11510974e-03j,\n 5.06302843e-02-1.75210735e-03j, 5.06042396e-02-2.38882773e-03j,\n 5.05701880e-02-3.02517013e-03j, 5.05281347e-02-3.66103387e-03j,\n 5.04780865e-02-4.29631832e-03j, 5.04200512e-02-4.93092298e-03j,\n 5.03540381e-02-5.56474743e-03j, 5.02800575e-02-6.19769139e-03j,\n 5.01981213e-02-6.82965469e-03j, 5.01082423e-02-7.46053735e-03j,\n 5.00104348e-02-8.09023955e-03j, 4.99047142e-02-8.71866165e-03j,\n 4.97910974e-02-9.34570421e-03j, 4.96696022e-02-9.97126802e-03j,\n 4.95402478e-02-1.05952541e-02j, 4.94030548e-02-1.12175637e-02j,\n 4.92580449e-02-1.18380984e-02j, 4.91052410e-02-1.24567600e-02j,\n 4.89446673e-02-1.30734505e-02j, 4.87763491e-02-1.36880725e-02j,\n 4.86003132e-02-1.43005286e-02j, 4.84165874e-02-1.49107220e-02j,\n 4.82252007e-02-1.55185561e-02j, 4.80261834e-02-1.61239348e-02j,\n 4.78195670e-02-1.67267621e-02j, 4.76053843e-02-1.73269429e-02j,\n 4.73836691e-02-1.79243820e-02j, 4.71544564e-02-1.85189850e-02j,\n 4.69177826e-02-1.91106578e-02j, 4.66736851e-02-1.96993067e-02j,\n 4.64222025e-02-2.02848387e-02j, 4.61633746e-02-2.08671610e-02j,\n 4.58972424e-02-2.14461816e-02j, 4.56238480e-02-2.20218088e-02j,\n 4.53432346e-02-2.25939515e-02j, 4.50554466e-02-2.31625192e-02j,\n 4.47605296e-02-2.37274220e-02j, 4.44585303e-02-2.42885704e-02j,\n 4.41494963e-02-2.48458757e-02j, 4.38334767e-02-2.53992497e-02j,\n 4.35105214e-02-2.59486048e-02j, 4.31806815e-02-2.64938541e-02j,\n 4.28440093e-02-2.70349114e-02j, 4.25005579e-02-2.75716909e-02j,\n 4.21503818e-02-2.81041079e-02j, 4.17935362e-02-2.86320780e-02j,\n 4.14300778e-02-2.91555178e-02j, 4.10600640e-02-2.96743443e-02j,\n 4.06835533e-02-3.01884755e-02j, 4.03006054e-02-3.06978300e-02j,\n 3.99112808e-02-3.12023273e-02j, 3.95156411e-02-3.17018875e-02j,\n 3.91137489e-02-3.21964316e-02j, 3.87056679e-02-3.26858813e-02j,\n 3.82914625e-02-3.31701592e-02j, 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0.77077077, 0.77277277, 0.77477477, 0.77677678, 0.77877878,\n 0.78078078, 0.78278278, 0.78478478, 0.78678679, 0.78878879,\n 0.79079079, 0.79279279, 0.79479479, 0.7967968 , 0.7987988 ,\n 0.8008008 , 0.8028028 , 0.8048048 , 0.80680681, 0.80880881,\n 0.81081081, 0.81281281, 0.81481481, 0.81681682, 0.81881882,\n 0.82082082, 0.82282282, 0.82482482, 0.82682683, 0.82882883,\n 0.83083083, 0.83283283, 0.83483483, 0.83683684, 0.83883884,\n 0.84084084, 0.84284284, 0.84484484, 0.84684685, 0.84884885,\n 0.85085085, 0.85285285, 0.85485485, 0.85685686, 0.85885886,\n 0.86086086, 0.86286286, 0.86486486, 0.86686687, 0.86886887,\n 0.87087087, 0.87287287, 0.87487487, 0.87687688, 0.87887888,\n 0.88088088, 0.88288288, 0.88488488, 0.88688689, 0.88888889,\n 0.89089089, 0.89289289, 0.89489489, 0.8968969 , 0.8988989 ,\n 0.9009009 , 0.9029029 , 0.9049049 , 0.90690691, 0.90890891,\n 0.91091091, 0.91291291, 0.91491491, 0.91691692, 0.91891892,\n 0.92092092, 0.92292292, 0.92492492, 0.92692693, 0.92892893,\n 0.93093093, 0.93293293, 0.93493493, 0.93693694, 0.93893894,\n 0.94094094, 0.94294294, 0.94494494, 0.94694695, 0.94894895,\n 0.95095095, 0.95295295, 0.95495495, 0.95695696, 0.95895896,\n 0.96096096, 0.96296296, 0.96496496, 0.96696697, 0.96896897,\n 0.97097097, 0.97297297, 0.97497497, 0.97697698, 0.97897898,\n 0.98098098, 0.98298298, 0.98498498, 0.98698699, 0.98898899,\n 0.99099099, 0.99299299, 0.99499499, 0.996997 , 0.998999 ,\n 1.001001 , 1.003003 , 1.00500501, 1.00700701, 1.00900901,\n 1.01101101, 1.01301301, 1.01501502, 1.01701702, 1.01901902,\n 1.02102102, 1.02302302, 1.02502503, 1.02702703, 1.02902903,\n 1.03103103, 1.03303303, 1.03503504, 1.03703704, 1.03903904,\n 1.04104104, 1.04304304, 1.04504505, 1.04704705, 1.04904905,\n 1.05105105, 1.05305305, 1.05505506, 1.05705706, 1.05905906,\n 1.06106106, 1.06306306, 1.06506507, 1.06706707, 1.06906907,\n 1.07107107, 1.07307307, 1.07507508, 1.07707708, 1.07907908,\n 1.08108108, 1.08308308, 1.08508509, 1.08708709, 1.08908909,\n 1.09109109, 1.09309309, 1.0950951 , 1.0970971 , 1.0990991 ,\n 1.1011011 , 1.1031031 , 1.10510511, 1.10710711, 1.10910911,\n 1.11111111, 1.11311311, 1.11511512, 1.11711712, 1.11911912,\n 1.12112112, 1.12312312, 1.12512513, 1.12712713, 1.12912913,\n 1.13113113, 1.13313313, 1.13513514, 1.13713714, 1.13913914,\n 1.14114114, 1.14314314, 1.14514515, 1.14714715, 1.14914915,\n 1.15115115, 1.15315315, 1.15515516, 1.15715716, 1.15915916,\n 1.16116116, 1.16316316, 1.16516517, 1.16716717, 1.16916917,\n 1.17117117, 1.17317317, 1.17517518, 1.17717718, 1.17917918,\n 1.18118118, 1.18318318, 1.18518519, 1.18718719, 1.18918919,\n 1.19119119, 1.19319319, 1.1951952 , 1.1971972 , 1.1991992 ,\n 1.2012012 , 1.2032032 , 1.20520521, 1.20720721, 1.20920921,\n 1.21121121, 1.21321321, 1.21521522, 1.21721722, 1.21921922,\n 1.22122122, 1.22322322, 1.22522523, 1.22722723, 1.22922923,\n 1.23123123, 1.23323323, 1.23523524, 1.23723724, 1.23923924,\n 1.24124124, 1.24324324, 1.24524525, 1.24724725, 1.24924925,\n 1.25125125, 1.25325325, 1.25525526, 1.25725726, 1.25925926,\n 1.26126126, 1.26326326, 1.26526527, 1.26726727, 1.26926927,\n 1.27127127, 1.27327327, 1.27527528, 1.27727728, 1.27927928,\n 1.28128128, 1.28328328, 1.28528529, 1.28728729, 1.28928929,\n 1.29129129, 1.29329329, 1.2952953 , 1.2972973 , 1.2992993 ,\n 1.3013013 , 1.3033033 , 1.30530531, 1.30730731, 1.30930931,\n 1.31131131, 1.31331331, 1.31531532, 1.31731732, 1.31931932,\n 1.32132132, 1.32332332, 1.32532533, 1.32732733, 1.32932933,\n 1.33133133, 1.33333333, 1.33533534, 1.33733734, 1.33933934,\n 1.34134134, 1.34334334, 1.34534535, 1.34734735, 1.34934935,\n 1.35135135, 1.35335335, 1.35535536, 1.35735736, 1.35935936,\n 1.36136136, 1.36336336, 1.36536537, 1.36736737, 1.36936937,\n 1.37137137, 1.37337337, 1.37537538, 1.37737738, 1.37937938,\n 1.38138138, 1.38338338, 1.38538539, 1.38738739, 1.38938939,\n 1.39139139, 1.39339339, 1.3953954 , 1.3973974 , 1.3993994 ,\n 1.4014014 , 1.4034034 , 1.40540541, 1.40740741, 1.40940941,\n 1.41141141, 1.41341341, 1.41541542, 1.41741742, 1.41941942,\n 1.42142142, 1.42342342, 1.42542543, 1.42742743, 1.42942943,\n 1.43143143, 1.43343343, 1.43543544, 1.43743744, 1.43943944,\n 1.44144144, 1.44344344, 1.44544545, 1.44744745, 1.44944945,\n 1.45145145, 1.45345345, 1.45545546, 1.45745746, 1.45945946,\n 1.46146146, 1.46346346, 1.46546547, 1.46746747, 1.46946947,\n 1.47147147, 1.47347347, 1.47547548, 1.47747748, 1.47947948,\n 1.48148148, 1.48348348, 1.48548549, 1.48748749, 1.48948949,\n 1.49149149, 1.49349349, 1.4954955 , 1.4974975 , 1.4994995 ,\n 1.5015015 , 1.5035035 , 1.50550551, 1.50750751, 1.50950951,\n 1.51151151, 1.51351351, 1.51551552, 1.51751752, 1.51951952,\n 1.52152152, 1.52352352, 1.52552553, 1.52752753, 1.52952953,\n 1.53153153, 1.53353353, 1.53553554, 1.53753754, 1.53953954,\n 1.54154154, 1.54354354, 1.54554555, 1.54754755, 1.54954955,\n 1.55155155, 1.55355355, 1.55555556, 1.55755756, 1.55955956,\n 1.56156156, 1.56356356, 1.56556557, 1.56756757, 1.56956957,\n 1.57157157, 1.57357357, 1.57557558, 1.57757758, 1.57957958,\n 1.58158158, 1.58358358, 1.58558559, 1.58758759, 1.58958959,\n 1.59159159, 1.59359359, 1.5955956 , 1.5975976 , 1.5995996 ,\n 1.6016016 , 1.6036036 , 1.60560561, 1.60760761, 1.60960961,\n 1.61161161, 1.61361361, 1.61561562, 1.61761762, 1.61961962,\n 1.62162162, 1.62362362, 1.62562563, 1.62762763, 1.62962963,\n 1.63163163, 1.63363363, 1.63563564, 1.63763764, 1.63963964,\n 1.64164164, 1.64364364, 1.64564565, 1.64764765, 1.64964965,\n 1.65165165, 1.65365365, 1.65565566, 1.65765766, 1.65965966,\n 1.66166166, 1.66366366, 1.66566567, 1.66766767, 1.66966967,\n 1.67167167, 1.67367367, 1.67567568, 1.67767768, 1.67967968,\n 1.68168168, 1.68368368, 1.68568569, 1.68768769, 1.68968969,\n 1.69169169, 1.69369369, 1.6956957 , 1.6976977 , 1.6996997 ,\n 1.7017017 , 1.7037037 , 1.70570571, 1.70770771, 1.70970971,\n 1.71171171, 1.71371371, 1.71571572, 1.71771772, 1.71971972,\n 1.72172172, 1.72372372, 1.72572573, 1.72772773, 1.72972973,\n 1.73173173, 1.73373373, 1.73573574, 1.73773774, 1.73973974,\n 1.74174174, 1.74374374, 1.74574575, 1.74774775, 1.74974975,\n 1.75175175, 1.75375375, 1.75575576, 1.75775776, 1.75975976,\n 1.76176176, 1.76376376, 1.76576577, 1.76776777, 1.76976977,\n 1.77177177, 1.77377377, 1.77577578, 1.77777778, 1.77977978,\n 1.78178178, 1.78378378, 1.78578579, 1.78778779, 1.78978979,\n 1.79179179, 1.79379379, 1.7957958 , 1.7977978 , 1.7997998 ,\n 1.8018018 , 1.8038038 , 1.80580581, 1.80780781, 1.80980981,\n 1.81181181, 1.81381381, 1.81581582, 1.81781782, 1.81981982,\n 1.82182182, 1.82382382, 1.82582583, 1.82782783, 1.82982983,\n 1.83183183, 1.83383383, 1.83583584, 1.83783784, 1.83983984,\n 1.84184184, 1.84384384, 1.84584585, 1.84784785, 1.84984985,\n 1.85185185, 1.85385385, 1.85585586, 1.85785786, 1.85985986,\n 1.86186186, 1.86386386, 1.86586587, 1.86786787, 1.86986987,\n 1.87187187, 1.87387387, 1.87587588, 1.87787788, 1.87987988,\n 1.88188188, 1.88388388, 1.88588589, 1.88788789, 1.88988989,\n 1.89189189, 1.89389389, 1.8958959 , 1.8978979 , 1.8998999 ,\n 1.9019019 , 1.9039039 , 1.90590591, 1.90790791, 1.90990991,\n 1.91191191, 1.91391391, 1.91591592, 1.91791792, 1.91991992,\n 1.92192192, 1.92392392, 1.92592593, 1.92792793, 1.92992993,\n 1.93193193, 1.93393393, 1.93593594, 1.93793794, 1.93993994,\n 1.94194194, 1.94394394, 1.94594595, 1.94794795, 1.94994995,\n 1.95195195, 1.95395395, 1.95595596, 1.95795796, 1.95995996,\n 1.96196196, 1.96396396, 1.96596597, 1.96796797, 1.96996997,\n 1.97197197, 1.97397397, 1.97597598, 1.97797798, 1.97997998,\n 1.98198198, 1.98398398, 1.98598599, 1.98798799, 1.98998999,\n 1.99199199, 1.99399399, 1.995996 , 1.997998 , 2. ]),\n array([ 0.00000000e+00, 1.95852809e-01, 3.66832565e-01, 4.93765595e-01,\n 5.67321172e-01, 5.89506039e-01, 5.72197109e-01, 5.33255344e-01,\n 4.91432512e-01, 4.61565735e-01, 4.51379322e-01, 4.60652255e-01,\n 4.82753430e-01, 5.07839643e-01, 5.26570502e-01, 5.33141782e-01,\n 5.26765596e-01, 5.11302134e-01, 4.93370831e-01, 4.79733190e-01,\n 4.74904893e-01, 4.79788730e-01, 4.91702428e-01, 5.05665383e-01,\n 5.16385339e-01, 5.20188219e-01, 5.16213633e-01, 5.06508322e-01,\n 4.95066391e-01, 4.86235815e-01, 4.83107830e-01, 4.86470949e-01,\n 4.94671022e-01, 5.04370994e-01, 5.11879353e-01, 5.14529784e-01,\n 5.11605594e-01, 5.04496332e-01, 4.96070954e-01, 4.89538900e-01,\n 4.87243960e-01, 4.89838130e-01, 4.96121351e-01, 5.03574536e-01,\n 5.09356791e-01, 5.11376986e-01, 5.09039679e-01, 5.03403051e-01,\n 4.96715149e-01, 4.91526300e-01, 4.89724667e-01, 4.91856633e-01,\n 4.96973911e-01, 5.03044342e-01, 5.07752300e-01, 5.09376031e-01,\n 5.07411745e-01, 5.02720302e-01, 4.97158072e-01, 4.92847352e-01,\n 4.91371152e-01, 4.93196132e-01, 4.97532487e-01, 5.02669586e-01,\n 5.06646957e-01, 5.07998908e-01, 5.06291318e-01, 5.02255222e-01,\n 4.97478619e-01, 4.93784697e-01, 4.92538758e-01, 4.94146244e-01,\n 4.97925443e-01, 5.02392838e-01, 5.05843036e-01, 5.06997513e-01,\n 5.05476247e-01, 5.01919049e-01, 4.97719476e-01, 4.94480860e-01,\n 4.93406050e-01, 4.94852421e-01, 4.98216124e-01, 5.02181766e-01,\n 5.05235179e-01, 5.06240012e-01, 5.04859173e-01, 5.01665368e-01,\n 4.97905547e-01, 4.95015416e-01, 4.94072503e-01, 4.95395647e-01,\n 4.98439316e-01, 5.02016895e-01, 5.04762162e-01, 5.05649928e-01,\n 5.04377848e-01, 5.01467591e-01, 4.98052273e-01, 4.95436242e-01,\n 4.94597876e-01, 4.95824558e-01, 4.98615689e-01, 5.01885834e-01,\n 5.04386009e-01, 5.05179891e-01, 5.03993723e-01, 5.01309411e-01,\n 4.98169702e-01, 4.95773838e-01, 4.95020202e-01, 4.96170100e-01,\n 4.98758281e-01, 5.01780355e-01, 5.04081945e-01, 5.04799016e-01,\n 5.03681670e-01, 5.01180275e-01, 4.98264634e-01, 4.96048532e-01,\n 4.95364813e-01, 4.96452885e-01, 4.98875721e-01, 5.01694793e-01,\n 5.03833145e-01, 5.04486336e-01, 5.03424626e-01, 5.01073060e-01,\n 4.98341820e-01, 4.96274364e-01, 4.95649206e-01, 4.96687154e-01,\n 4.98973945e-01, 5.01625138e-01, 5.03627804e-01, 5.04227141e-01,\n 5.03210617e-01, 5.00982784e-01, 4.98404665e-01, 4.96461324e-01,\n 4.95885833e-01, 4.96883047e-01, 4.99057165e-01, 5.01568486e-01,\n 5.03457418e-01, 5.04010829e-01, 5.03031004e-01, 5.00905856e-01,\n 4.98455656e-01, 4.96616689e-01, 4.96083770e-01, 4.97047964e-01,\n 4.99128457e-01, 5.01522697e-01, 5.03315732e-01, 5.03829591e-01,\n 5.02879414e-01, 5.00839630e-01, 4.98496639e-01, 4.96745860e-01,\n 4.96249765e-01, 4.97187417e-01, 4.99190115e-01, 5.01486175e-01,\n 5.03198065e-01, 5.03677572e-01, 5.02751060e-01, 5.00782105e-01,\n 4.98528999e-01, 4.96852904e-01, 4.96388915e-01, 4.97305579e-01,\n 4.99243888e-01, 5.01457720e-01, 5.03100874e-01, 5.03550324e-01,\n 5.02642293e-01, 5.00731748e-01, 4.98553773e-01, 4.96940916e-01,\n 4.96505110e-01, 4.97405652e-01, 4.99291129e-01, 5.01436432e-01,\n 5.03021452e-01, 5.03444435e-01, 5.02550299e-01, 5.00687357e-01,\n 4.98571742e-01, 4.97012263e-01, 4.96601347e-01, 4.97490112e-01,\n 4.99332906e-01, 5.01421640e-01, 5.02957730e-01, 5.03357275e-01,\n 5.02472892e-01, 5.00647982e-01, 4.98583478e-01, 4.97068752e-01,\n 4.96679934e-01, 4.97560888e-01, 4.99370069e-01, 5.01412858e-01,\n 5.02908133e-01, 5.03286820e-01, 5.02408373e-01, 5.00612859e-01,\n 4.98589389e-01, 4.97111752e-01, 4.96742643e-01, 4.97619481e-01,\n 4.99403307e-01, 5.01409751e-01, 5.02871478e-01, 5.03231522e-01,\n 5.02355418e-01, 5.00581366e-01, 4.98589743e-01, 4.97142274e-01,\n 4.96790817e-01, 4.97667051e-01, 4.99433184e-01, 5.01412113e-01,\n 5.02846911e-01, 5.03190227e-01, 5.02313013e-01, 5.00552990e-01,\n 4.98584688e-01, 4.97161032e-01, 4.96825440e-01, 4.97704480e-01,\n 4.99460165e-01, 5.01419852e-01, 5.02833850e-01, 5.03162107e-01,\n 5.02280396e-01, 5.00527305e-01, 4.98574261e-01, 4.97168480e-01,\n 4.96847193e-01, 4.97732419e-01, 4.99484643e-01, 5.01432984e-01,\n 5.02831959e-01, 5.03146618e-01, 5.02257025e-01, 5.00503952e-01,\n 4.98558395e-01, 4.97164840e-01, 4.96856487e-01, 4.97751312e-01,\n 4.99506948e-01, 5.01451628e-01, 5.02841129e-01, 5.03143479e-01,\n 5.02242545e-01, 5.00482624e-01, 4.98536916e-01, 4.97150112e-01,\n 4.96853480e-01, 4.97761421e-01, 4.99527365e-01, 5.01476013e-01,\n 5.02861470e-01, 5.03152652e-01, 5.02236784e-01, 5.00463055e-01,\n 4.98509540e-01, 4.97124076e-01, 4.96838089e-01, 4.97762835e-01,\n 4.99546143e-01, 5.01506480e-01, 5.02893313e-01, 5.03174344e-01,\n 5.02239738e-01, 5.00445013e-01, 4.98475863e-01, 4.97086284e-01,\n 4.96809981e-01, 4.97755470e-01, 4.99563502e-01, 5.01543503e-01,\n 5.02937229e-01, 5.03209018e-01, 5.02251580e-01, 5.00428288e-01,\n 4.98435341e-01, 4.97036038e-01, 4.96768557e-01, 4.97739063e-01,\n 4.99579641e-01, 5.01587700e-01, 5.02994049e-01, 5.03257415e-01,\n 5.02272666e-01, 5.00412688e-01, 4.98387270e-01, 4.96972358e-01,\n 4.96712923e-01, 4.97713158e-01, 4.99594747e-01, 5.01639869e-01,\n 5.03064915e-01, 5.03320595e-01, 5.02303560e-01, 5.00398031e-01,\n 4.98330748e-01, 4.96893932e-01, 4.96641839e-01, 4.97677075e-01,\n 4.99608999e-01, 5.01701028e-01, 5.03151331e-01, 5.03399994e-01,\n 5.02345064e-01, 5.00384140e-01, 4.98264631e-01, 4.96799040e-01,\n 4.96553653e-01, 4.97629876e-01, 4.99622576e-01, 5.01772466e-01,\n 5.03255253e-01, 5.03497507e-01, 5.02398271e-01, 5.00370830e-01,\n 4.98187465e-01, 4.96685457e-01, 4.96446197e-01, 4.97570300e-01,\n 4.99635667e-01, 5.01855822e-01, 5.03379210e-01, 5.03615609e-01,\n 5.02464630e-01, 5.00357907e-01, 4.98097397e-01, 4.96550312e-01,\n 4.96316648e-01, 4.97496683e-01, 4.99648478e-01, 5.01953195e-01,\n 5.03526470e-01, 5.03757520e-01, 5.02546050e-01, 5.00345146e-01,\n 4.97992044e-01, 4.96389879e-01, 4.96161328e-01, 4.97406837e-01,\n 4.99661254e-01, 5.02067297e-01, 5.03701285e-01, 5.03927449e-01,\n 5.02645040e-01, 5.00332281e-01, 4.97868311e-01, 4.96199293e-01,\n 4.95975411e-01, 4.97297877e-01, 4.99674292e-01, 5.02201675e-01,\n 5.03909243e-01, 5.04130944e-01, 5.02764924e-01, 5.00318974e-01,\n 4.97722117e-01, 4.95972116e-01, 4.95752499e-01, 4.97165965e-01,\n 4.99687981e-01, 5.02361043e-01, 5.04157787e-01, 5.04375410e-01,\n 5.02910148e-01, 5.00304773e-01, 4.97547990e-01, 4.95699698e-01,\n 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\n", 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\u001b[0;36m1\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1j\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpi\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1j\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpi\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 19\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m+=\u001b[0m\u001b[0mx_term\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 20\u001b[0m \u001b[0miplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mgo\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mScatter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreal\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 21\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mTypeError\u001b[0m: Cannot cast ufunc add output from dtype('complex128') to dtype('float64') with casting rule 'same_kind'"]}]}}, "f0cd9a8ba3e14b3b848f222f9c0d4465": {"model_module": "@jupyter-widgets/controls", "model_module_version": "1.2.0", "model_name": "IntSliderModel", "state": {"description": "k_max", "layout": "IPY_MODEL_7b51ec05f15d41e79cc092a75d88f0ae", "max": 10, "min": 1, "style": 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